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HW4-solution - Solution to HW4 4.5 by dene X = the dierence...

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Solution to HW4 April 29, 2010 4.5 by define X = the difference between # of heads and # of tails in flips. When n heads, 0 tails: X=n; when n-1 heads, 1 tails: X=n-2; · · · · · · ; when 0 heads, n tails: X=-n; so we conclude that X can take n possible values in { n - 2 i, for i from 0 to n } . 4.28 Fist of all, define r.v X=# of defective items in the sample; and X follows hyper-geometric distribution with parameters N=20, m=4, and n=3. By formula on page 162, E [ X ] = nm N = 4 * 3 20 = 3 5 . 4.35 Define X=amount of money we win; because X does follow any famous distribution, we have to find its possible values and associated probabilities. Possible values= { 1.1 ,-1 } . P(X=1.1)=P(2 marbles of the same color)= ( 5 2 ) + ( 5 2 ) ( 10 2 ) = 0 . 4444444; P(X=-1)=1-P(X=1.1)= 0.5555556; E[X]=1 . 1 * 0 . 4444444 - 1 * 0 . 5555556 = - 0 . 06666676. Var[X]=0 . 444444 * (1 . 1 + 0 . 06666676) 2 + 0 . 5555556 * ( - 1 + 0 . 06666676) 2 = 1 . 088888. 4.40 Define X=# of correct out of 5 problems; and we know that the probability of correctly answer one problem= 1 3 , since their 3 possible answer for each of them and we answer by guessing.
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